# Liouville theorems for a family of very degenerate elliptic non linear operators

## TL;DR

This paper establishes Liouville-type nonexistence theorems for nonnegative solutions of certain fully nonlinear degenerate elliptic equations involving operators defined by eigenvalues of the Hessian, revealing different behaviors based on operator degeneracy.

## Contribution

It provides new Liouville theorems for operators ${\

## Key findings

- Liouville theorems analogous to Laplace in certain cases
- Distinct results due to operator degeneracy
- Nonexistence of solutions under specified conditions

## Abstract

We prove nonexistence results of Liouville type for nonnegative viscosity solutions of some equations involving the fully nonlinear degenerate elliptic operators ${\cal P}^\pm_k$, defined respectively as the sum of the largest and the smallest $k$ eigenvalues of the Hessian matrix. For the operator ${\cal P}^+_k$ we obtain results analogous to those which hold for the Laplace operator in space dimension $k$. Whereas, owing to the stronger degeneracy of the operator ${\cal P}^-_k$, we get totally different results.